#202831

order	512
nilpotency class	3
isoclinism	isoclinic class of #19328
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,15)(8,14)(9,11)(10,13)(12,16)(17,20)(18,19)(21,22,26,28)(23,29,31,36)(24,25,32,33)(27,34,35,30);(1,3,4,2)(6,11)(12,19)(13,14)(15,20)(21,23,25,30,26,31,33,34)(22,27,24,29,28,35,32,36);(5,7,9,17)(6,12,11,19)(8,16,10,18)(13,15,14,20)(21,24,25,28,26,32,33,22)(23,29,30,35,31,36,34,27);(5,8,9,10)(6,13,11,14)(7,16,17,18)(12,15,19,20)(22,28)(24,32)(27,35)(29,36)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=e,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^c=dh,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110027);